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EXTENSIONS of ORDERED GROUPS and SEQUENCE COMPLETION(I) by CHARLES HOLLAND^)

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EXTENSIONS of ORDERED GROUPS and SEQUENCE COMPLETION(I) by CHARLES HOLLAND^) EXTENSIONS OF ORDERED GROUPS AND SEQUENCE COMPLETION(i) BY CHARLES HOLLAND^) 1. Introduction. In this paper the relationships between certain extensions of totally (linearly) ordered groups (o-groups) are studied. The additive notation is employed although the groups are not, in general, assumed to be abelian. The four types of extensions considered are defined as follows : Let G S H be o-groups. (a) H is an a-extension of G if for every 0 < he H there exists ge G and a positive integer n such that h ^ g ^ nh. (b) H is a b-extension of G if for every 0 < n e H there exists g e G and a positive integer n such that g ^ n :£ ng. (c) H is a c-extension of G if for every 0 < h e Ü there exists g e G such that for all integers n, n(h —g) < h. (t) H is a t-extension of G if for every h,h',h"eH with h < h' < h" there exists g e G such that h < g < h". If xe {a, b,c,f}, G is x-closed if G has no proper x-extensions. The connect- ing notion between the different types of extensions is that of sequences in o- groups; of particular importance are cauchy sequences (definition in §2) and pseudo sequences (definition in §3). Several authors have discussed similar con- cepts, but in different or more special cases. Cohen and Goffman [3 ; 4] consider cauchy sequences in abelian groups; Everett and Ulam [8] consider countable cauchy sequences; Banaschewski [1] deals with cauchy filters in partially ordered groups; Gravett [9] relates pseudo sequences to c-extensions in divisible abelian groups. §2 contains definitions and a basic lemma concerning sequences. It is also shown that every r-extension and every c-extension is a ¿-extension, and that every b-extension is an a-extension. Some results concerning r-extensions and cauchy sequences ("j" theorems) are stated without proof. In §3 some theorems ("C" theorems) concerning pseudo sequences and c-extensions, including ana- logues to the "T" theorems, are proved. In §4 are stated the "B" theorems concerning sequences and b-extensions with proofs only sketched, as the proofs Received by the editors May 14, 1962. i1) This paper is based on material in the author's Ph.D. thesis, written under the direc- tion of P. F. Conrad, to whom the author expresses sincerest gratitude. (2) The author holds a NATO Postdoctoral Fellowship at the University of Tübingen. 71 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 72 CHARLES HOLLAND [April are mostly similar to those in §3. In §6 are examples which illustrate the limita- tion of the theory. The system of numbering is such that Theorems Ti, Ci, and Bi are analogues. 2. Cauchy sequences and i-extensions. Throughout this paper G will denote an o-group. The pairs (Gy,Gy) of convex subgroups of G such that Gy covers Gy are called convex pairs. Gy is normal in Gy and Gy/Gy is o-isomorphic to a sub- group of R, the additive group of real numbers. The groups Gy/Gy are called the components of G. If A and B are subsets of G, A < B means every element of A is less than every element of B. For g e G, | g | is the max of g and —g. A sequence (in G) is a collection {sx | a e A} of elements of G such that A is a well-ordered set with no final element. The breadth of a sequence {sx} is the inter- section of all convex subgroups C of G satisfying: (1) C is covered by a convex subgroup C. (2) For each geC \C there exists peA such that if a, ß > p then sx — sßeC and sx- sß + C <\g\ + C Note that the breadth is a convex subgroup. Here and in several places later on the definition is "one-sided" due to lack of commutativity. This would more appropriately be called "left breadth," the "right breadth" being defined sim- ilarly. The introduction of this complexity seems, however, to be of no par- ticular aid. An element s of G is a limit of the sequence {sa} if for every convex pair (C,C) such that C contains the breadth of {sx} and for every geC \C, there exists v e A such that if a > v, sx — seC and \sx — s\ + C <\g\ + C. We say {sx} converges to s. Lemma 1. If r and s are limits of {sx} then r —s lies in the breadth of {sx}. Proof. By way of contradiction, if r —s is not in the breadth B then r-seC'\C where (C,C) is a convex pair and BçzC. Now for all sufficiently large a, 2\sx — r\ + C<\r — s\ + C. For C'/C is o-isomorphic to a subgroup of the real numbers, and if C'/C is not discrete then there exists g e G such that 2|g| + C<|r — s| + C and by assumption, for all a large enough, Is,, — r\ + C < \g\ + C. If C'/C is discrete it is cyclic generated by some f + C; then for all a large enough | sx - r | + C < \f | + C so that sx — reC, in which case 2\sx- r\ + C = C <\r - s\ + C. Choose a so that sx-r, sx — seC and 2\ sx - r\+ C <\r - s\ + C and 2\ sx - s\ + C <\ r - s\ + C. Then \r-s\ + C = \r-sa + sx-s\ + C ^ \sa-r\ + \s„-s\+C <\r - s\ + C,& contradiction. Hence r —seB. Lemma 2. Let H be an a-extension of G. If {gx} is a sequence of elements of G, if N is the breadth of {gx} in G, and if B is the breadth of {gx}in H, then BnG = N. If geG is a limit of {gx} in G then g is a limit of {gx} in H. Proof. (K,K') is a convex pair of if if and only if (K n G, K' C\ G) is a con- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1963] ORDERED GROUPS AND SEQUENCE COMPLETION 73 vex pair of G. In the definition of breadth, it is required that the differences gx — gß become arbitrarily small in certain components K'/K and K' n G/K n G, respectively. As K' n G/K nG Z K'/K are subgroups of the real numbers, the differences in question become small in K' nG/XnG if and only if they be- beco me small in K'/K. A similar remark holds for the differences gx — g. A sequence {sx} is a left cauchy sequence if its breadth is 0. {sx} is a cauchy sequence if both {sx} and { —sa} are left cauchy sequences. Example 1 of §6 contains a left cauchy sequence which is not a cauchy sequence. In the topology generated by the open intervals, every o-group is a topological group. It is easily seen from the definition that if is a f-extension of G if and only if G is dense in H in the order topology. Banaschewski [1] has shown that the topological completion t(G) of an o-group G is an o-group which extends the order of G. Banaschewski's proof, as well as the earlier and less general ones of Cohen and Goffman [3] and Everett and Ulam [8], proceeds by completing G via "dedekind cuts." It is also possible to construct t(G) by choosing cauchy sequences of appropriate length, showing that they form an o-group under term- wise addition, and continuing in a manner analogous to the cauchy sequence construction of the real numbers. The proof of this, from which the following results easily follow, is notationally involved and sufficiently similar to the clas- sical results that the details are omitted here. Theorem Tl. If H is a t-extension of G and heH\G then there is a cauchy sequence of elements of G with limit h (and with no limits in G). Corollary T2. If every cauchy sequence in G has a limit then G is t-closed. Let T be an ordered set, and for each y e T let Ry be a subgroup of the real numbers. The group H(T,Ry) consisting of those functions in the large direct sum of the Ry whose support is inversely well ordered, is called a Hahn group, and is ordered by calling a function positive if it is positive on the largest element of its support. Lemma T4. If H(T,Ry) is a Hahn group and either T has no smallest ele- ment or y0 is the smallest element of V and Ryo is the reals or the integers, then H(T,Ry) is t-closed. Proof. Let {fx} be a cauchy sequence in H. It follows from the definition of cauchy sequence that for each y e T except the smallest element of T, if any, for all sufficiently large a and ß, fx(y) =fß(y) =f(y), say. Moreover, if T has a smallest element y0 then the sequence {fx(y0)} is a cauchy sequence in the ordinary sense in R, and hence converges to/(y0), say. It is easy to verify that the function / is an element of H and the limit of {/„}. Theorem T6. G is t-closed if and only if every cauchy sequence in G con- verges.
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